Disorder-induced topology in quench dynamics
DDisorder-induced topology in quench dynamics
Hsiu-Chuan Hsu,
1, 2, ∗ Pok-Man Chiu, and Po-Yao Chang † Graduate Institute of Applied Physics, National Chengchi University, Taipei 11605, Taiwan Department of Computer Science, National Chengchi University, Taipei 11605, Taiwan Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan (Dated: January 28, 2021)We study the effect of strong disorder on topology and entanglement in quench dynamics. Although disorder-induced topological phases have been well studied in equilibrium, the disorder-induced topology in quenchdynamics has not been explored. In this work, we predict a disorder-induced topology of post-quench statescharacterized by the quantized dynamical Chern number and the crossings in the entanglement spectrum in (1 +1) dimensions. The dynamical Chern number undergoes transitions from zero to unity, and back to zero whenincreasing disorder strength. The boundaries between different dynamical Chern numbers are determined bydelocalized critical points in the post-quench Hamiltonian with the strong disorder. An experimental realizationin quantum walks is discussed.
INTRODUCTION
Topological phases of matter out-of-equilibrium and theirphase transitions have attracted much theoretical and exper-imental interest. Their topological and non-equilibrium fea-tures have been demonstrated in various systems includingultracold-atomic gases [1–4], quantum optics [5–7], super-conducting qubits [8, 9], and condensed matter systems [10–18]. Among these, the topological Floquet systems havebeen widely studied [19–21]. These systems exhibit protectedboundary states which are robust in the presence of disorder.More recently, topological phases in dynamical quench sys-tems are proposed [22–26]. For example, for a trivial stateunder a sudden quench by the Su-Schrieffer-Heeger (SSH)model, the topology of the post-quench state is characterizedby the dynamical Chern numbers [22, 24]. The quantization ofdynamical Chern number describes a skyrmion texture of thepost-quench pseudospin in the momentum-time space [27].The topology in quench dynamics has been shown experimen-tally in photonic quantum walks [28–30] and superconductingqubits [31, 32]. Moreover, the entanglement spectrum pro-vides an additional probe of the topology of quench dynam-ics. The robustness of crossings in the entanglement spectrumof the post-quench states indicates the nontrivial topology inquench dynamics [23, 24].Besides the topological structures that emerge in quenchdynamics, non-trivial topology can arise from disordered sys-tems in equilibrium. In the strong disorder regime, an unex-pected topological phase with extensive boundary states is sta-bilized by the strong disorder. This phase is termed “topolog-ical Anderson insulator” [33–37] and the transition betweentrivial and non-trivial phases is described by the delocalizationcriticality[38]. A generalization of the topological Andersoninsulator to Floquet systems is proposed [39–42]. The strongdisorder drives trivial Floquet systems into topological phasesthat host chiral edge modes coexisting with the localized bulkstates in two-dimensional lattices. The transition also links tothe localization-delocalization transition [43].Although there are extensive studies in disorder-inducedtopology in Floquet systems, the effect of disorder on topol- ogy in quench dynamics is less discussed. It is shown thatthe crossings in the entanglement spectrum are robust againstdisorder and interactions [23]. However, it has not knownwhether disorder could induce topology in quench dynamics.In this work, we demonstrate the disorder-induced topologyin quench dynamics. We consider a quench protocol describedby a trivial state under a sudden quench by the SSH Hamil-tonian in the presence of strong disorder. The topology ofthe post-quench state is characterized by the dynamical Chernnumber and is zero (unity) when the SSH model is trivial(non-trivial). We start at the clean limit where the post-quenchstate carrying a vanishing dynamical Chern number. Whenthe disorder strength above the critical value, the post-quenchstate has a quantized dynamical Chern number. The entangle-ment spectrum of the post-quench states shows robust cross-ings which indicate the disorder-induced topology in quenchdynamics. The post-quench SSH model in this strong disorderregime has a disorder-induced winding number. We identifythe phase boundaries of the SSH model from the localizationlength. The phase boundaries coincide with the transitionsbetween vanishing and non-vanishing dynamical Chern num-bers of the post-quench states. Our results demonstrate thatthe disorder-induced topology in quench dynamics in (1 + 1) dimensions is directly related to the topological Anderson in-sulator in dimension. THE QUENCH PROTOCOL
We consider an eigenstate | Ψ (cid:105) of pre-quench Hamilto-nian H pre at t = 0 under a sudden quench by a post-quenchHamiltonian H . The post-quench state is governed by theunitary evolution | Ψ( t ) (cid:105) = exp[ − iHt ] | Ψ (cid:105) . In the follow-ing, we consider the post-quench Hamiltonian with two parts H = H + H U . H is the SSH Hamiltonian H = N x (cid:88) i =0 J c † i,a c i,b + J c † i +1 ,a c i,b + h . c ., (1)where i is the label of the unit-cell, N x is the total num-ber of the unit-cell, c † ia ( b ) , c ia ( b ) are the creation and anni- a r X i v : . [ c ond - m a t . d i s - nn ] J a n hilation operators on sublattices a, b on the i th unit cell. J denotes the intracell coupling, while J denotes the intercellcoupling. In the clean limit, one can do Fourier transformto the momentum space, H = (cid:80) k ψ † ( k ) H ( k ) ψ ( k ) with ψ ( k ) T = ( c a ( k ) , c b ( k )) T . The single-particle SSH Hamilto-nian H ( k ) is written as H ( k ) = h x ( k ) σ x + h y ( k ) σ y , where h x ( k ) = J + J cos k , h y ( k ) = J sin k . Here σ x,y are Paulimatrices and act on the sublattices a, b . The lattice constantis taken to be unity. The energy is E ± = ± (cid:113) h x + h y . For J /J > ( J /J < ), the winding number of the groundstate is W = 1(0) . The single-particle Hamiltonian is be-long to the BDI symmetry class with the time-reversal sym-metry T H ( k ) T − = H ( − k ) and the particle-hole symme-try CH ( k ) C − = −H ( − k ) . Here T = K and C = σ z K with K being the complex conjugation.The second part in the post-quench Hamiltonian is time-reversal and particle-hole symmetry preserving disorder H U = N (cid:88) i =1 U i c † i,a c i,b + U i c † i,a c i +1 ,b + h . c ., (2)where U i is the random intracell(intercell) couplingstrength given by the random number in the uniform distribu-tion (cid:2) − W / , W / (cid:3) . The topological phase of the post-quench Hamiltonian H + H U is characterized by the windingnumber and the disorder-induced transition of the post-quenchHamiltonian is accompanied by delocalization [38]. Thus, thephase boundary can be obtained by the divergence of local-ization length. By solving the Schr¨odinger equation for zeroenergy with the transfer matrix method, one obtains the local-ization length λ = (cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ n ln n (cid:89) i =1 (cid:12)(cid:12)(cid:12)(cid:12) J ,i J ,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3)where J ,i = J + U i and J ,i = J + U i . More details canbe found in the supplemental material [44]. It has been shownthat an anlytical expression for the localization length can beobtained [38] λ = (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:12)(cid:12) J + W (cid:12)(cid:12) J W + (cid:12)(cid:12) J − W (cid:12)(cid:12) J W − (cid:12)(cid:12) J − W (cid:12)(cid:12) J W − (cid:12)(cid:12) J + W (cid:12)(cid:12) J W + (cid:12)(cid:12)(cid:12)(cid:12) . (4)The phase diagram given by Eq. (4) is shown in Fig. 1(a). For J = 1 . , as denoted by the white dashed line in the diagram,the two delocalization points are at W = 1 . and . . For . ≤ W ≤ . , the system becomes a topological Andersoninsulator [45].We consider the initial single particle-state being the prod-uct state | ψ i (cid:105) = (1 , T ⊗ | i (cid:105) , where (1 , T denotes oneparticle at site a , | i (cid:105) = (0 , · · · , , · · · , T denotes theonly non-vanishing i -th element with i being label of thesite i = 1 . . . N x . The corresponding pre-quench single-particle Hamiltonian is H pre . = − σ z . In non-interactingcase, each single-particle state individually evolves underthe single-particle Hamiltonian | ψ i ( t ) (cid:105) = exp[ − i H t ] | ψ i (cid:105) . The post-quench many-particle state is expressed in terms ofanti-symmetrizing the products of post-quench single-particlestates, | Ψ( t ) (cid:105) = (cid:80) i j (cid:15) i i ··· i Nx (cid:78) i j | ψ i j ( t ) (cid:105) . Here (cid:15) i i ··· i Nx is the Levi-Civita symbol.Due to the chiral symmetry ( S = T C ), the single-particleHamiltonian H is homotopically equivalent to the flat-bandHamiltonian H F constructed from the projectors H F = | ψ + (cid:105)(cid:104) ψ + | − | ψ − (cid:105)(cid:104) ψ − | , where | ψ ± (cid:105) are the eigenstates of H with positive (negative) energies. Since the topology doesnot change under the flattening process, we consider thepost-quench state for the flat-band Hamiltonian, | ψ ( t ) (cid:105) = e − i H F t | ψ (cid:105) . In the following, the flat-band Hamiltonian is uti-lized unless otherwise stated.The effective Hamiltonian can be constructed by an unitaryrotation H eff . ( k, t ) = e − i H F t σ z e i H F t [23] or from the pseu-dospin presentation [24], H eff ( k, t ) = h y ( k ) sin 2 t √ h x ( k ) + h y ( k ) σ x − h x ( k ) sin 2 t √ h x ( k ) + h y ( k ) σ y + cos 2 tσ z . Since we choose the ini-tial state that explicitly breaks the particle-hole symmetry,the effective Hamiltonian does not preserve the particle-hole symmetry. However, in addition to the time-reversalsymmetry T H eff ( k, t ) T − = H eff ( − k, − t ) , there are twotwo-fold symmetries σ z H eff ( k, t ) σ z = H eff ( k, − t ) and σ x H eff ( k, t ) σ x = −H eff ( − k, t ) . These two additional sym-metries together with the time-reversal symmetry lead to a Z classification in (1 + 1) dimensions [44]. The former two-fold symmetry acts like the reflection symmetry in the timedomain. There are two fixed points t = 0 and π/ such that [ σ z , H eff ( k, σ z , H eff ( k, π/ . The dynamicalChern number in this effective Hamiltonian is quantized inthe half of the momentum-time space k ∈ [0 , π ] , t ∈ [0 , π/ [46–49]. BERRY PHASE AND DYNAMICAL CHERN NUMBER
To determine the dynamical Chern number in the real space,we compute the Berry phase with the twisted boundary con-dition [50–52] by the overlap matrix proposed in [53, 54].The overlap matrix at a given time t is defined as M (cid:96)ij ( t ) = (cid:104) ψ θ (cid:96) i ( t ) | ψ θ (cid:96) +1 j ( t ) (cid:105) , where i, j denotes the index of the single-particle state, θ (cid:96) denotes the twisted boundary phase in dis-crete form θ (cid:96) = π(cid:96)L [52], where L is the number of meshpoints for the twisted boundary condition, l = 1 , · · · , L . TheBerry phase is given by γ ( t ) = Im (cid:34) ln det L (cid:89) (cid:96) =1 M (cid:96) ( t ) (cid:35) . (5)The Berry phase as a function of t (Berry phase flow) has nojump when the post-quench state is trivial [Fig.1 blue dots].On the other hand, when the post-quench state is topological,the Berry phase flow has π jumps at t = π/ as shown bythe red dots in Fig.1(b).The dynamical Chern number is obtained by integrating the FIG. 1. (a) The phase diagram for the the disordered SSH Hamil-tonian H = H + H U determined by the localization length usingEq. (4). The white dashed line denotes J = 1 . . (b) The time-dependent Berry phase in the clean limit. The blue dots are for thetrivial post-quench state with no Berry phase flow ( J /J = 1 . ).The red dots are for the topological post-quench state with a Berryphase flow from t = 0 to t = π/ ( J /J = 0 . ). time derivative of the Berry phase C dyn = 12 π (cid:90) π/ dt ∂γ ( t ) ∂t . (6)The quantization of the dynamical Chern number corre-sponds to the mapping of the space-time torus to the Blochsphere of the pseudospin ˆ n i ( t ) = (cid:104) ψ i ( t ) | (cid:126)σ | ψ i ( t ) (cid:105) . Since the π/ is the half period of the pseudospin precession from thenorth pole to the south pole, integrating out all the space isequivalent to counting the numbers of the pseudospin wrap-ping around the entire Bloch sphere[24].The disorder-induced dynamical Chern number is shown bythe red dots in Fig. 2(a). In the case with disorder strengths W = 2 W = W , we find for . (cid:46) W (cid:46) . , the dy-namical Chern number is close to an integer with vanishingfluctuations.The classification of the topology of quench dynamics de-pends on the initial state and the post-quench Hamiltonian[23, 24]. We investigate the relation between the quantizeddynamical Chern number and the disordered SSH model bycomputing the localization length with Eq. (3). In the nu-merical calculation, n is taken to be that ensures conver-gence.The result is shown by the blue dots in Fig. 2. Thedelocalization happens at W = 1 . and . which coin-cide with the transition points (half-integer) of the dynamicalChern number of the post-quench state. Therefore, the quan-tized dynamical Chern number is attributed to the disorder-induced topology of the post-quench Hamiltonian. ENTANGLEMENT SPECTRUM
The entanglement spectrum provides additional informa-tion on the topology induced by disorder in quench dynamics.It is shown that the crossings in the entanglement spectrumreveal the topological properties in both the equilibrium sys-tems [55–61] and out-of-equilibrium systems[23, 24, 62, 63].The presence (absence) of the robust crossings in the entan-glement spectrum indicates the post-quench state is topolog-ical (trivial). To compute the entanglement properties, the
FIG. 2. The disorder-average dynamical Chern number (DCN) andthe localization length for the post-quench Hamiltonian. The errorbar is the standard deviation and is very small in this scale. There aremore than disorder realizations for each data point. The parame-ters are J = 1 . , J = 1 , N x = 400 . system is bipartite spatially into A and B subsystems, wherethe post-quench many-body state is expressed as | Ψ( t ) (cid:105) = (cid:80) i,j C ij ( t ) | A i (cid:105)| B j (cid:105) with | A ( B ) i (cid:105) being the local basis insubsystem A ( B ) . We can compute the reduced density ma-trix ρ A ( t ) = Tr B | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | = N e − H A ( t ) , where H A ( t ) isreferred to the entanglement Hamiltonian, N is the normaliza-tion constant, and the spectrum of H A ( t ) is the entanglementspectrum.In free-fermion systems, the eigenvalues of the reduceddensity matrix can be obtained from the correlation matrix C x , x (cid:48) ( t ) = (cid:104) Ψ( t ) | c † x c x (cid:48) | Ψ( t ) (cid:105) = (cid:80) i | ψ i ( x (cid:48) , t ) (cid:105)(cid:104) ψ i ( x , t ) | ,where | ψ i ( x , t ) (cid:105) is the postquench single-particle state. Thespectrum ξ ( t ) of the correlation matrix C x , x (cid:48) ( t ) with x, x (cid:48) be-ing restricted in A is related to the entanglement spectrum (cid:15) ( t ) by ξ ( t ) = 1 / (1 + e (cid:15) ( t ) ) [64]. For simplicity, we refer ξ ( t ) tothe entanglement spectrum in the following discussion.As shown in Fig. 3(a), in the clean limit at J /J = 1 . , thepost-quench state is trivial and no crossings in the entangle-ment spectrum. When the disorder strength is above the criti-cal values, the entanglement spectrum of the post-quench stateshows a crossing at t = π/ [Fig. 3(b)]. The existence of thecrossings in the entanglement spectrum agrees with the non-vanishing dynamical Chern number of the post-quench state.This indicates that the disorder-induced topology in quenchdynamics can be detected from the entanglement spectrum. EXPERIMENTAL REALIZATION
Discrete-time quantum walks are great platforms for sim-ulating the topological phases of matter[28, 65, 66], quan-tum quenches[29, 30], and disorder phenomena[67–69]. Fol-lowing Ref. [29], the discrete-time evolution operator fora one-dimensional lattice with single photons can be engi-neered by the cascaded half-wave plates and beam displac-ers. The Hilbert space is spanned by the polarization states (cid:1) (cid:1) t0.51 (cid:1) (a) (b) (cid:1) (cid:1) t0.51 (cid:1) FIG. 3. The entanglement spectrum of the postquench state with thebipartition l A = l B = N x / , where l A ( B ) is the length of the sub-system A ( B ) and N x is the length of the total system. The parame-ters are J = 1 . , J = 1 . (a) W = 0 (clean limit). (b) W = 3 .There are disorder realizations for each data point. {| P + (cid:105) , | P − (cid:105)} and the position state | x (cid:105) with x ∈ Z . Thecorresponding evolution operator for each time step is U = R ( φ / SR ( φ ) SR ( φ / , where R ( φ ) rotate the polariza-tion by φ with respect to y -axis, and S is the shift operator S = (cid:80) x | x − (cid:105)(cid:104) x | ⊗ | P + (cid:105)(cid:104) P + | + | x + 1 (cid:105)(cid:104) x | ⊗ | P − (cid:105)(cid:104) P − | . Thepolarization angle φ ( x ) is spatially dependent and disordercan be introduced by choosing different φ ( x ) for different x . In a translation-invariant case, the unitary operator is diag-onal in momentum space and the effective Hamiltonian hasthe form H eff ( k ) = − i ln U ( k ) = E ( k ) (cid:126)n ( k ) · (cid:126)σ . It is shownthat this quantum walk protocol [29] can simulate a suddenquench between H ieff ( k ) and H feff ( k ) of the SSH model. Here H ieff ( k ) and H feff ( k ) refer to the pre-quench and the post-quench Hamiltonians.The topological property of the post-quench state can beextrapolated from the post-quench pseudospin n ( k, t ) =Tr[ ρ ( k, t ) σ ] with ρ ( k, t ) = | ψ k ( t ) (cid:105)(cid:104) ψ k ( t ) | . The post-quenchpseudospin n ( k, t ) can be measure directly in the momentum-time space. In a disorder-free case ( J J , W J ) = (0 . , , thepseudospin winds π clockwise on the x − y plane from k = 0 to k = 2 π [see the green line in Fig. 4(a)]. At t = 0 and t = π/ , pseudospin is polarized along + z and − z direc-tions. The pseudospin texture in k ∈ [0 2 π ] and t ∈ [0 , π/ isa Skyrmion, i. e., the core of the Skyrmion is pseudospin po-larized along + z direction which corresponds to t = 0 and theouter contour of the Skyrmion is pseudospin polarized along − z direction which corresponds to t = π/ . In the experi-mental setup, the Hamiltonian is not flattened and the period-icity of the pseudospin is no longer π for different momenta k . Due to the robustness of topology, the Skyrmion texture isstill robust as shown in Fig. 4(b).However, in the presence of disorder, the momen-tum is no longer a good quantum number. Nev-ertheless, we define a disorder-average density ma-trix in the pseudomomentum-time space ρ (cid:48) (˜ k, t ) = (cid:80) i =0 (cid:80) x ,x e − i ˜ k ( x − x ) (cid:104) ψ x ( t ) | σ i | ψ x ( t ) (cid:105) σ i . Here · · · denotes the disorder average. We compute the normalizedpseudospin texture in the pseudomomentum-time space inthe strong disorder region[70]. We observe a Skyrmion-like (a) (b)(c)
We predict the disorder-induced topology in quench dy-namics in (1+1) dimensions. The topology is characterizedby the dynamical Chern number and crossings in the entan-glement spectrum. We show the boundaries between trivialand nontrivial post-quench states are identified by delocalizedcritical points in the post-quench Hamiltonian. The quantizeddynamical Chern number in (1+1) dimensions corresponds tothe winding number of the one-dimensional topological An-derson insulating phase of the SSH model. Finally, we pro-pose this phenomenon can be realized in quantum walk ex-periments.
ACKNOWLEDGMENTS
The authors thank Ching-Hao Chang and Chao-ChengKaun for hosting the workshop of quantum materials at Re-search Center for Applied Sciences, Academia Sinica, wherethe work was partially initiated. H.C.H. was supported bythe Ministry of Science and Technology (MOST) in Taiwan,MOST 108-2112-M-004-002-MY2. P. -M. C and P.-Y.C.were supported by the Young Scholar Fellowship Program un-der MOST Grant for the Einstein Program MOST 109-2636-M-007-003-. ∗ [email protected] † [email protected][1] M. S. Foster, V. Gurarie, M. Dzero, and E. A. Yuzbashyan,Phys. Rev. Lett. , 076403 (2014).[2] K. Plekhanov, G. Roux, and K. Le Hur, Phys. Rev. B ,045102 (2017).[3] N. R. Cooper, J. Dalibard, and I. B. Spielman, Rev. Mod. Phys. , 015005 (2019).[4] G. Salerno, H. M. Price, M. Lebrat, S. H¨ausler, T. Esslinger,L. Corman, J.-P. Brantut, and N. Goldman, Phys. Rev. X ,041001 (2019).[5] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podol-sky, F. Dreisow, S. Nolte, M. Segev, and A. 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In this document we present detail derivations on the symmetry analysis, entanglement spectrum, the Wannier center follow,and the derivation of the localization length.
SYMMETRY ANALYSIS AND TOPOLOGICAL CLASSIFICATION
The effective Hamiltonian we discussed in the main text is constructed as H eff ( k, t ) = e − i H ( k ) t H ( k ) e − i H ( k ) t with H ( k )( H ( k )) being the pre (post)-quench Hamiltonian. The effective Hamiltonian has the following symmetries T H eff ( k, t ) T − = H eff ( − k, − t ) , R t H eff ( k, t ) R − t = H eff ( k, − t ) , M x H eff ( k, t ) M − x = −H eff ( − k, t ) , (7)where T = R t = M x = 1 , {R t , M x } = 0 , and [ T , R t ] = [ T , M x ] = 0 .The effective Hamiltonian can be expressed in terms of the effective massive Dirac Hamiltonian H eff ( k, t ) = kγ + tγ + M γ , with { γ i , γ j } = 0 ( i = 0 , , ). We construct the minimal effective Dirac Hamiltonian in terms of the tensor productform of the Pauli matrices γ = σ x ⊗ σ x , γ = σ y ⊗ I × , γ = σ z ⊗ I × , T = I × ⊗ σ z K , R t = σ z ⊗ σ z , M x = σ x ⊗ I × . (8)One can check the only allowed mass term which preserving all the symmetries is the γ . For the Z classification, we needto make copies of the original effective Hamiltonian. For simplicity, we just make one copy. The double Hamiltonian is H eff ( k, t ) = kγ ⊗ I × + tγ ⊗ I × + M γ ⊗ I × , for which there are no other symmetry-preserving mass terms. Thisindicates that different phases are not adiabatically connected in this system. On the other hand, we can flip one momentum ofthe copy and construct the double Hamiltonian, H eff ( k, t ) = kγ ⊗ σ z + tγ ⊗ I × + M γ ⊗ I × . There is another symmetryallowed mass term (anti-commute with γ ⊗ I × ⊗ I × ), M = σ y ⊗ σ y ⊗ σ y . This indicates the system are all in the samephases. We conclude from the above analysis that the system belongs to a Z classification. Similar classification schemes can befound in Refs. [46–49] CORRELATION FUNCTION FORMALISM IN QUENCH SETUPS
We consider an initial state contains N particles. Each single-particle state we denote by | φ α ( x ) (cid:105) , α = 1 , · · · , N , x is theinternal degrees of freedom, including position, spin, and the band. We require these single-particle states are orthonormal, (cid:80) x (cid:104) φ α ( x ) | φ β ( x ) (cid:105) = δ α,β . The N-particle initial state can be expressed as the Slater determinant of these single-particle state, | Ψ (cid:105) = Det[ | φ i ( x j ) (cid:105) ] , i, j = 1 , · · · , N. (9)We consider a unitary evolution of this initial state | Ψ (cid:105) by a static Hamiltonian H = (cid:80) x , x (cid:48) H x , x (cid:48) c † x c x (cid:48) , where c ( † ) x is theannihilation (creation) operator. Each single-particle state under this evolution is | φ α ( x , t ) (cid:105) = (cid:80) x (cid:48) exp[ − i H x , x (cid:48) t ] | φ α ( x (cid:48) ) (cid:105) , α =1 , · · · , N . The postquench N-particle state is | Ψ( t ) (cid:105) = e − iHt | Ψ (cid:105) = Det[ | φ i ( x j , t ) (cid:105) ] = (cid:89) i d † i ( t ) | (cid:105) , (10)where d † i ( t ) = e − iHt d † i e iHt = e − iHt (cid:88) y V i y c † y e iHt = (cid:88) x , y V i y U y , x ( t ) c † x (11)with U y , x ( t ) = e − i H y , x t and V i y being a unitary matrix that rotates d † i to c † y .The postquench single-particle state is d † i ( t ) | (cid:105) = (cid:88) x , y V i y U y , x ( t ) c † x | (cid:105) == (cid:88) x | φ i ( x , t ) (cid:105) . (12) (cid:1) t0.10.20.30.40.5wc (cid:1) t0.10.20.30.40.5wc (a) (b) FIG. 5. The Wannier center as a function of t . (a) Disorder-free Hamiltonian ( J J , W J ) = (1 . , , (b) disordered Hamiltonian ( J J , W J ) =(1 . , . There are disorder realizations. The correlation function constructed from the N-particle postquench state is C x , x (cid:48) ( t ) = (cid:104) Ψ( t ) | c † x c x (cid:48) | Ψ( t ) (cid:105) = (cid:104) | (cid:89) α d α e iHt c † x c x (cid:48) e − iHt (cid:89) β d † β | (cid:105)(cid:105) = (cid:104) | (cid:89) α d α [ (cid:88) y ,i d i U x , y ( t ) V y i ] † [ (cid:88) y (cid:48) ,j U x (cid:48) , y (cid:48) ( t ) V y (cid:48) j d i ] (cid:89) β d † β || (cid:105) = (cid:88) i [ (cid:88) y U x , y ( t ) V y i ] † [ (cid:88) y (cid:48) U x (cid:48) , y (cid:48) ( t ) V y (cid:48) i ]= (cid:88) i | φ i ( x (cid:48) , t ) (cid:105)(cid:104) φ i ( x , t ) | . (13)The correlation matrix can be used for computing the entanglement spectrum. The existence of the ESCs can detect thetopology of the postquench state, as we will demonstrate in several examples. WANNIER CENTER WITH DISORDERS
In translational invariant systems, the Wannier orbits are constructed from the Bloch states u n k ( r ) , w n ( r − R ) = (cid:82) d k e i k · ( r − R ) u n k ( r ) , with Ω being the volume of the system, R is the position of the unit-cell, and r is the local posi-tion of the Wannier orbits within the unit-cell. In an insulator, these Wannier orbits are localized state and are the eigenstateof the projected position operator X P = P XP , where P is the projector to the occupied states which are well defined in aninsulator.To construct the Wannier orbits without using Bloch states, we first write down the Hamiltonian in the real space H IJ , where I ( J ) includes the band indices and positions. The spectrum can exhibit a gap and the corresponding occupied states | ψ αI (cid:105) arewell defined. Here α is the eigen-energy index. The corresponding projectors is P IJ = (cid:80) α ∈ occ . | ψ αI (cid:105)(cid:104) ψ αJ | . The positionoperator can be defined by as a diagonal matrix diag(1 , · · · , , , · · · , · · · , N, · · · , N ) , where N is the total number of sitesand at each site there are L bands. The projected position operator can be constructed as usual X P = P XP .Since the Wannier orbits are the eigenstates of the X P , we can find diagonalize the X P and get the set of eigenstates. If the setof the eigenstates are localized states, then these states are the Wannier orbits and the corresponding eigenvalues are the positionof the Wannier states. The Wannier center of a localized state in M -th site can be defined as wc M = |(cid:104) w M | X P | w M (cid:105) − M | . Wehave < (cid:104) x M (cid:105) < . We can further define the average Wannier center wc = N (cid:80) Nm =1 wc M . In the present of chiral symmetryin one dimension gapped systems, the average Wannier center can have two values wc = 0 and . . The former corresponds toa trivial phase [Fig. 5(a)] and the later is the topological phase [Fig. 5(b)] . LOCALIZATION LENGTH
When electrons are localized, the wave function exponentially decays with length, i.e. φ L ∝ e − L/λ , where φ L = (cid:80) Ln =1 ( φ na , φ nb ) T c † n is the eigenstate of the Hamiltonian H = H o + H U with length n , φ na/b are the coefficients for sub-lattice a/b at site n and λ is the localization length. The Schrodinger equation for zero eigenenergy state becomes ( J + U n ) φ nb + ( J + U n ) φ n − ,b = 0 , (14) ( J + U n ) φ na + ( J + U n ) φ n +1 ,a = 0 . (15)The above equations give the ratio of coefficients between the first and the last site, (cid:12)(cid:12) φ La (cid:12)(cid:12) = (cid:81) Ln =1 (cid:12)(cid:12) J + U n J + U n φ a (cid:12)(cid:12) and (cid:12)(cid:12) φ Lb (cid:12)(cid:12) = (cid:81) Ln =1 (cid:12)(cid:12) J + U n J + U n φ b (cid:12)(cid:12) for each sublattice, respectively. The final localization length for the system is the minimum of that of thesublattices. Thus, the localization length is given by λ = 1 L ln L (cid:89) n =1 (cid:12)(cid:12) J + U n J + U n (cid:12)(cid:12) . (16)The equation can be solved analytically [38], as shown in Eq.4.Another approach to calculate the localization length is via Green’s function. The localization length λ is defined by λ = − lim L →∞ L Tr ln | G ,L | , (17)where n is the total number of site of the one-dimensional Hamiltonian, G ,L is the propagator connecting the first and the lastslice of the system [71]. G ,n is computed with the iterative Green’s function method [71–73] by computing the onsite Green’sfunction G n,n = ( E − h n − U f G n − ,n − U b ) and G ,n = G ,n − U b G n,n recursively till n is large enough for convergence,where h n = ( J + U ,n ) σ x , U f ( b ) = ( J + U n )( σ x + ( − ) iσ y ) / and U n are defined in Eq. 2. Within this method, theHamiltonian is constructed in a slicing scheme, i.e. H N = N (cid:88) i =1 ( | i (cid:105) h i (cid:104) i | + | i (cid:105) U b (cid:104) i + 1 | + | i + 1 (cid:105) U f (cid:104) i | ) (18)for the system with N slices, where | i (cid:105) is the state for the i -th slice, U f ( b ) is the forward (backward) hopping matrices betweenthe neighboring slices, andTo calculate the Greens function for the system with N + 1 slices, the Hamiltonian for N + 1 slices is H N +1 = H N + | N + 1 (cid:105) h N +1 (cid:104) N + 1 | + H (cid:48) , (19)where h N +1 is the Hamiltonian for N +1 -th slices, the hopping matrix H (cid:48) = | N (cid:105) U b (cid:104) N +1 | + | N +1 (cid:105) U f (cid:104) N | between the N − thand N + 1 − th slice is treated as a perturbing term to H N + | N + 1 (cid:105) h N +1 (cid:104) N + 1 | . According to Dyson equation, the perturbedGreens function is given by G N +1 = G o + G o H (cid:48) G N +1 , where G o = G N + | N + 1 (cid:105) ( E − h N +1 ) − (cid:104) N + 1 | . Substitute H (cid:48) into the Dyson equation, one obtains the Greens function for N + 1 slices ( G N +1 ) in which the submatrices are given by (cid:104) N + 1 | G N +1 | N + 1 (cid:105) = ( E − h N +1 − U f (cid:104) N | G N | N (cid:105) U b ) − , (20) (cid:104) | G N +1 | N + 1 (cid:105) = (cid:104) | G N | N (cid:105) U b (cid:104) N + 1 | G N +1 | N + 1 (cid:105) . (21)Eq. 20 and 21 are the main iterative equations for obtaining the localization length shown in Fig. 7(a). OTHER PARAMETERS FOR THE DISORDER-INDUCED TOPOLOGY IN QUENCH DYNAMICS
In the clean limit, the dynamical Chern numbers are calculated for ≤ J ≤ and J = 1 of the SSH Hamiltonian H o [Eq.(1).] The results are shown in Fig. 6. For J > , the static Hamiltonian becomes trivial and the dynamical Chern number iszero.We consider the case with vanishing intercell disorder W = 0 . We find for . (cid:46) W (cid:46) . , the dynamical Chern number isclose to an integer with vanishing fluctuations as shown in Fig. 7(a). The phase boundaries, where the dynamical Chern numberis close to half-integer, are at W = 1 . , . . The localization length λ also indicates delocalized transitions at the same valuesof W [Fig. 7 (a)]. The entanglement spectrum has a crossing at t = π/ when the post-quench state has integer dynamicalChern number W = 3 [Fig. 7(b)].0 FIG. 6. The dynamical Chern number (DCN) for the static Hamiltonian in the clean limit. The parameters are J = 1 . , J = 1 , N x = 400 . π π t0.51 ξ (a) (b) FIG. 7. (color online) (a) The disorder averaged mean dynamical Chern number and localization length obtained from Eq.17 for the quenchHamiltonian. The error bar is the standard deviation. The parameters are J = 1 . , J = 1 , N x = 100 , L = 400 . (b) The entanglementspectrum of the post-quench state with W = 3 . The parameters are J = 1 . , J = 1 , N x = 400 . There are more than50